Operator $K$-theory algebra spectra of $C^*$-algebras

TL;DR

This paper constructs algebra spectra representing operator $K$-theory of $C^*$-algebras, introducing a new graded ring structure that reflects the multiplicative properties of topological $K$-theory.

Contribution

It develops a novel framework of $ extit{L}$-permutative categories and constructs $E_$-ring spectra representing operator $K$-theory, generalizing previous algebraic structures.

Findings

Introduces commutative algebra spectra for operator $K$-theory.

Defines $ extit{L}$-permutative categories with coherent multiplicative actions.

Constructs $E_$-ring spectra from categories of projections and partial isometries.

Abstract

We construct commutative algebra spectra that represent the operator $K$-theory of $C^*$-algebras, which are algebras over the commutative ring spectra that represent topological $K$-theory. The spectral multiplicative structure introduces a new graded commutative ring structure on the $K$-groups, generalizing the well-known graded ring structure of commutative $C^*$-algebras. This last structure reflects the multiplicative structure of topological $K$-theory via Gelfand duality, Swan's theorem and the fiber tensor product. We introduce $\mathscr L$-permutative categories, a generalization of bipermutative categories, which are permutative categories equipped with a multiplicative structures induced by coherent actions of the linear isometries operad. The main class of examples of interest are categories whose objects are projection matrices of the unitization of the stabilizations $\widetilde{\mathfrak{KA}}$ of a $C^*$-algebras $\mathfrak A$, and morphisms partial isometries witnessing the Murray-von Neumann relation. We then construct $E_\infty$-ring spaces out of them by adapting the usual method applied to bipermutative categories. The delooping functor of the recognition principle, the homotopical augmentation ideal and localization at the Bott element then give us our algebra spectra.